Surjectivity for finite sets of combinators by weak reduction

Abstract

A set F ≡ {F ₁,...F _t} of closed λ-terms in β-η-normal form (combinators) is s-t-surjective iff the system of equations F _i X ₁...X _s = y_i (i=1,...,t), where the y_i's are arbitrary variables, is solvable using only weak (β) reduction. It is well known that F is s-t surjective if s is sufficiently large.

In this paper sufficient conditions are given for the s-t-surjectivity of F if s=n, where n is the maximum number of initial abstractions among its elements. An interesting consequence is that F is unconditionally (n+1)-t-surjective. Another technical result which revealed itself useful for the proof of the main theorem is: if every element of F has the same number n of initial abstractions, then there are ∞ⁿ ways of finding a set L ≡ {L ₁,...L _t} of combinators such that, for i=1,...,t and any given λ-terms Y₁,..., Y_t, F _i X₁...X_n = Y_i is solvable ⇔ L _i X=Y_i is solvable.

This research has been supported by grants of Ministero della Pubblica Istruzione, Italia.

PhD student in Computer Science.